Dineutron-dineutron correlation in ⁸He

The existence of proton deficient nuclear systems has been a subject of debate in nuclear physics. In this context, the tetra-neutron system, comprising four neutrons, has garnered significant interest for decades [1, 2, 3, 4, 5, 6]. A crucial characteristic of the tetra-neutron system is that in the lowest energy configuration, one of the two-neutron pairs, known as dineutron ($2n$), occupies the $s$ state, while the other pair must occupy the $p$ state due to the Pauli exclusion principle. Consequently, the four-neutron correlation may be weakened in a vacuum as the two-$2n$ systems should occupy distinctly different orbits with each other.

The two-$2n$ correlations may be enhanced when a core nucleus, such as a ⁴He nucleus, is introduced into the four-neutron system. This is because, in the ⁸He nucleus, four neutrons are bound around the ⁴He core nucleus, and all four can occupy the $p$ state as their lowest energy configuration. As a result, the four-neutron correlation could emerge under a democratic situation, in contrast to the tetra-neutron system in a vacuum. It is important to note that in the ⁷H (${}^{3}{\rm H}+4n$) system, the four neutrons are no longer bound, although numerous experimental and theoretical investigations have focused on exploring the resonance states of ⁷H [7, 8, 9, 10, 11, 12, 13, 14].

The $2n$ correlations in the ⁸He nucleus have been theoretically discussed in Refs. [15, 16, 17], even though the neutron number of six in ⁷H and ⁸He corresponds to the subclosure of the $p_{3/2}$ orbits in the nuclear shell model, where the spin-orbit interaction acts attractively. The spin-orbit interaction serves as a driving force for promoting the independent particle motion of each nucleon with good total angular momentum [18] and leads to the breaking of the spin-singlet $2n$ clusters. However, if the four valence neutrons in the $p$ shell have extended spatial distributions outside the interaction region of the spin-orbit force, it would be possible to form the $2n$ clusters there. Recent experimental results support such spatially extended four-neutron wave functions: The four neutrons in the ⁸He nucleus are bound only by 3.11 MeV from the ${}^{4}{\rm He}+4n$ threshold energy [19], and the matter radius is evaluated as 2.53(2) fm with a neutron skin thickness of 0.82(2) fm, implying an extended four-neutron distribution [20].

We note that the $2n$-$2n$ correlation has not been investigated sufficiently, although the $2n$ correlations are extremely important in weakly bound systems and have been widely discussed in neutron-rich nuclei [21, 22, 23]. In Refs. [15, 17], the squared overlap between the ground-state wave function of ⁸He and two $2n$ cluster configurations was examined. In those studies, two $2n$ clusters were isotopically distributed around ⁸He by means of the Tohsaki-Horiuchi-Schuck-Röpke (THSR) wave function [24]. A similar approach was adopted in Ref. [25]. Several ab initio calculations were performed for the ⁸He nucleus. Many physical properties can be interpreted in connection with the realistic interactions but unfortunately the $2n$-$2n$ correlation was not discussed there [26, 27, 28, 29]. Ref. [16] conducted the Hartree-Fock-Bogoliubov calculations with the Wood-Saxon potential, where the four-body density of the valence neutrons and the $2n$-$2n$ correlation were discussed but that obtained nuclear radius is unrealistically larger than the experimental value. An investigation using a realistic wave function is needed to extract an amount of the $2n$-$2n$ correlation quantitatively.

The recent experiment posited that the ground state of ⁸He is strongly deformed [30]. This suggests that the ⁸He configuration is not a simple subclosure configuration of $p_{3/2}$. Instead, it may exhibit enhanced four-neutron correlations. Moreover, the possibility of the so-called soft dipole mode in ⁸He was recently pointed out in Ref. [31], implying the formation of a four-neutron cluster around the ⁴He core.

The purpose of this paper is to investigate the $2n$-$2n$ correlation in the ⁸He nucleus. We perform microscopic cluster model calculations that explicitly include various geometric configurations of two-$2n$ clusters. We remark that a strong dineutron correlation was implied within a ⁴He core plus $4n$ picture [32]. We take special care to describe the valence four-neutron wave functions around the ⁴He core by superposing numerous configurations. To verify our wave function, we calculate the proton-⁸He elastic scattering and total reaction cross sections and compare them with available experimental data. Finally, we discuss whether the four neutrons occupying the $p$ states can favorably form two $2n$ clusters under a democratic condition and the relationship between the two-$2n$ correlations and nuclear deformation.

Here, we introduce a microscopic ⁴He+$4n$ model, wherein the wave function is fully antisymmetrized. We construct basis states with different neutron configurations and superpose them based on the generator coordinate method (GCM) after the angular momentum projection [33]. The coefficients of each basis state are determined by diagonalizing the norm and Hamiltonian matrices. Using the obtained ground-state wave function of ⁸He, we calculate the one-body density distributions for the cross-section calculations with the Glauber model [34].

The Hamiltonian $(H)$ consists of kinetic energy $(t_{i})$ and potential energy ($v_{ij}$) terms as

$$H=\sum_{i=1}^{8}t_{i}-T_{\rm c.m.}+\sum_{i<j}v_{ij},$$

(1)

where the center-of-mass kinetic energy $T_{\rm c.m.}$ is subtracted to guarantee the translation-invariance of the wave functions. For the potential part, the interaction consists of the central, spin-orbit, and Coulomb terms. We employ the Volkov No.2 [35] effective nucleon-nucleon interaction with the Majorana exchange parameter of $M=0.6$, which is known to reproduce the low-energy scattering phase shift of ${}^{4}{\rm He}+^{4}$He [36]. The original Volkov interaction does not have the Heisenberg and Bartlett exchange terms, but we introduce $B=H=0.07$ just as in Ref. [13] to remove the spurious bound state of the two-neutron system. For the spin-orbit part, we use the spin-orbit term of the G3RS interaction [37], which is a realistic interaction originally developed to reproduce the nucleon-nucleon scattering phase shifts. The strength of the spin-orbit interaction is set to 2000 MeV, reproducing the low-energy scattering phase shift of $p+^{4}$He [37].

The wave function of ⁸He is described by a superposition of fully antisymmetrized ($\mathcal{A}$) ${}^{4}{\rm He}(\alpha)+4n$ wave function as

$\displaystyle\Phi=\mathcal{A}\left\{\Phi_{\alpha}(\bm{R}_{\alpha})\Phi_{4n}(\bm{R}_{1},\bm{R}_{2},\bm{R}_{3},\bm{R}_{4})\right\},$

(2)

where the $\alpha$ and $4n$ wave functions, $\Phi_{\alpha}$ and $\Phi_{4n}$, are expressed by the product of the single-particle wave function with a Gaussian form as used in many other cluster models, including the Brink model [33]

$$\phi^{\tau,\sigma}\left(\bm{r}\right)=\left(\frac{2\nu}{\pi}\right)^{\frac{3}{4}}\exp\left[-\nu\left(\bm{r}-\bm{R}\right)^{2}\right]\chi^{\tau,\sigma},$$

(3)

where $\bm{R}$ is a Gaussian center parameter related to the expectation value for the position of the nucleon, and $\chi^{\tau,\sigma}$ are the spin and isospin parts of the wave function. For the size parameter $\nu$, we here use $\nu=0.255$ fm^-2, slightly smaller than the value for the free ⁴He nucleus, which is reasonable, as the ⁴He core swells due to the interaction and Pauli principle from the valence neutrons [38, 39]. The $\alpha$ cluster as the core nucleus can be expressed by four nucleons with the spin and isospin saturated configuration sharing the same $\bm{R}_{\alpha}$ value. Similarly, the $2n$ cluster can also be expressed by two neutrons with spin-up and spin-down with $\bm{D}_{1}\equiv\bm{R}_{1}=\bm{R}_{2}$ and $\bm{D}_{2}\equiv\bm{R}_{3}=\bm{R}_{4}$.

Each Slater determinant is projected to the eigenstates of parity and angular momentum by using the projection operator

$$P_{J^{\pi}M}^{K}=P^{\pi}\frac{2J+1}{8\pi^{2}}\int d\Omega\,\left\{D_{MK}^{J}\right\}^{*}R\left(\Omega\right).$$

(4)

Here ${D_{MK}^{J}}$ is the Wigner $D$-function and $R\left(\Omega\right)$ is the rotation operator acting on the spatial and spin parts of the wave function. This integration over the Euler angle $\Omega$ is numerically performed. The operator $P^{\pi}$ is for the parity projection. Here we take $P^{+}=\left(1+P^{r}\right)/\sqrt{2}$ for the positive-parity states, where $P^{r}$ is the parity-inversion operator.

The generated many different Slater determinants ($\{P_{J^{\pi}M}^{K}\left|\Phi_{i}\right>\}$) are superposed based on GCM [33]. After normalizing each basis state, the total wave function $\Psi_{J^{\pi}M}$ is written as

$$\Psi_{J^{\pi}M}=\sum_{i}c_{i}P_{J^{\pi}M}^{K}\left|\Phi_{i}\right>$$

(5)

The coefficients $\left\{c_{i}\right\}$ for a linear combination of the Slater determinants are obtained together with the energy eigenvalue $E$ when we diagonalize the norm and Hamiltonian matrices, namely by solving the Hill-Wheeler equation [40]

	$\displaystyle\sum_{j}\left[\left<\Phi_{i}\right|(P_{J^{\pi}M}^{K})^{\dagger}HP_{J^{\pi}M}^{K}\left|\Phi_{j}\right>\right.$
	$\displaystyle\left.-E\left<\Phi_{i}\right|(P_{J^{\pi}M}^{K})^{\dagger}P_{J^{\pi}M}^{K}\left|\Phi_{j}\right>\right]c_{j}=0.$		(6)

In this study, we prepare ${}^{4}{\rm He}+2n+2n$ wave functions with various isosceles triangle configurations and randomly generated ${}^{4}{\rm He}+4n$ (the $\alpha$ core + four free neutrons) wave functions.

The geometric configurations of the ${}^{4}{\rm He}+2n+2n$ wave functions are the isosceles triangles where the two sides of the ⁴He-$2n$ distance $D\equiv|\bm{D}1-\bm{R}_{\alpha}|=|\bm{D}_{2}-\bm{R}_{\alpha}|$ are taken as $D=1$, 2, 3, 4, 5, and 6 fm, and its opening angle $\Theta$ is taken as $\Theta=30^{\circ}$, 60^∘, $90^{\circ}$, $120^{\circ}$, and $150^{\circ}$ with the spin saturated configuration. Figure 1 shows the calculated potential energy surface of $J^{\pi}=0^{+}$ for the ${}^{4}{\rm He}+2n+2n$ configuration as a function of $D$ and $\Theta$. We can see a local energy minimum of $E=-17.97$ MeV at around $D=3.0$ fm and $\Theta=80^{\circ}$. Note that the rms distance of the two neutrons in the $2n$ cluster is 2.43 fm, which implies developed $2n$ clusters in the surface region in the ground state of ⁸He. Large positive values of the potential energy surface come from the Pauli principle.

For the $4n$ configurations, we randomly generate the Gaussian center parameters $\bm{R}_{1},\bm{R}_{2},\bm{R}_{3}$, and $\bm{R}_{4}$ of the four neutrons, where two of them are spin-up and the remaining two are spin-down. To accelerate the energy convergence, these random numbers are generated by following an exponential distribution having a width of 1.4 fm for the distance between neutrons with the up and down spins and the distance between their center position and the $\alpha$ core. The center of mass of each Slater determinant is shifted to the origin before the superposition.

Before discussing the ground state of ⁸He, we calculate ⁶He by randomly generated two-neutron configurations around an $\alpha$ cluster to verify our choice of the model Hamiltonian. The binding energy is found to be $-1.33$ MeV from the $\alpha+n+n$ threshold with 200 basis states. The value is slightly overbinding compared with the experimental data of $-0.975$ MeV [41] but is within the acceptable range because the calculated total reaction cross section on a carbon target is 719 mb at 800 MeV/nucleon, which is in good agreement with the experimental interaction cross section at 790 MeV/nucleon, 722$\pm 6$ mb [42]. The details of the cross-section calculation will be described for a ⁸He case later. The root-mean-square (rms) matter radius is calculated as 2.44 fm, which is slightly smaller than the empirical value 2.48$\pm$0.03 fm [43].

Figure 2 draws the energy convergence of the $0^{+}$ state of ⁸He as a function of the number of basis states superposed. The first 30 basis states correspond to the ${}^{4}{\rm He}+2n+2n$ basis states, and the subsequent 1,100 basis states represent the ⁴He+$4n$ basis states. To achieve convergence, more than 1,000 basis states are required.

The calculated energy of the ⁸He nucleus is found to be $-3.14$ MeV from the threshold, which is in reasonable agreement with experimental data, $-3.10$ MeV [41]. This result demonstrates that the wave function used in this study provides a good description of the four-valence neutrons of the ⁸He nucleus.

Figure 3 displays the point-matter, proton, and neutron density distribution of ⁸He. We see more extended neutron density distribution than proton one, indicating a thick neutron skin. The rms point-proton and matter radii of these density distributions are calculated as 1.82 fm and 2.50 fm, respectively. The calculated point-proton radius is in good agreement with the experimental ones deduced from the charge radius, 1.80(3) fm [44] and 1.81(3) fm [45].

To further verify our wave function, we calculate the total interaction cross sections on a carbon target as well as proton-nucleus elastic scattering cross sections. These calculations are performed based on the Glauber model [34] with the optical limit approximation [34, 46]. The nucleon-target formalism [47] is employed for the total reaction cross section calculation on a carbon target. The inputs to the theory are the one-body density distributions and the profile function which describes the nucleon-nucleon scattering properties. The profile function [48] has been well tested as shown in many examples of proton-nucleus [49, 50, 51] and nucleus-nucleus reactions [52, 53, 49, 54, 55, 56, 57]. For more details, see, for example, Refs. [58, 59] and references therein, showing the most recent application of this model. The essential input for the cross-section calculations, the one-body density distribution of the ground state of ⁸He, is obtained after solving the Hill-Wheeler equation (Eq. (6)). The density distribution is calculated in the angular momentum projected space, which is free from the center of mass motion, and the detail is described in Ref. [60].

Using these density distributions, the total reaction cross section on a carbon target at an incident energy of 800 MeV/nucleon is calculated as 798 mb. This value is slightly smaller than the experimental cross section of 817(6) mb at 790 MeV/nucleon [42] but still within the acceptable range considering the uncertainties of previous observations [45].

Figure 4 compares the calculated differential cross section for proton-⁸He scattering at an incident proton energy of 680 MeV with available experimental data at low [61] and high [62] four momentum transfer regions. Although there are some deviations at intermediate momentum transfer regions, the overall agreement between theory and experiment is achieved. To see it more quantitatively at low four momentum transfer regions, in Fig. 4 (b), we also plot the cross sections in a linear scale. This good agreement confirms the validity of the wave function used in this study.

Using the realistic wave function obtained in this way, the two-$2n$ correlation in ⁸He is investigated by analyzing the wave function components. The squared overlap between the ground state of ⁸He and the ⁴He+$2n$+$2n$ cluster configuration with isosceles triangular configurations is evaluated. Figure 5 displays the squared overlap results, showing that a ⁴He-$2n$ distance of $\sim 3$ fm gives the largest squared overlap and a peak structure is found around the $2n$-⁴He-$2n$ opening angle of $\Theta=80^{\circ}$. This large overlap area corresponds to a deep pocket in the potential energy surface coming from the triangular cluster configuration (Fig. 1).

The value of squared overlap at the peak position is about 0.45, indicating that ⁸He contains a significant amount of the two-$2n$ component with a large ⁴He-$2n$ distance. Despite the neutron number 6 corresponding to the closure of $p_{3/2}$ in the $jj$-coupling shell model, the spin-orbit interaction does not completely break the $2n$ clusters due to the weakly bound nature of the system; neutrons also stay beyond the interaction range of the spin-orbit interaction, $\sim 2.5$ fm [37]. The rapid drop of the squared overlap beyond $\Theta=90^{\circ}$ indicates that obtuse triangular shapes are not favored, which also suggests the correlation between the two $2n$ clusters. We also compute the squared overlap between the $0\hbar\omega$ harmonic-oscillator configuration [$(0s_{1/2})^{4}(0p_{3/2})^{4}$] and the ground state wave function. This overlap value is 0.40, which is comparable to the two-$2n$ component. We remark that the coupling with the continuum states significantly enhances the component of the continuum $p_{3/2}$ orbits [63].

To relate the two-$2n$ component with the nuclear shape, it is instructive to quantify the degree of deformation for each basis. The $\beta$ parameters is given by  [64]

$\displaystyle\beta$

$\displaystyle=\sqrt{a_{0}^{2}+2a_{2}^{2}},$

(7)

where the dimensionless deformation parameter is defined by

$$a_{m}=\frac{4\pi}{5}\frac{Q_{2m}}{AR^{2}}$$

(8)

with $R=1.2A^{1/3}$ fm and the quadrupole deformation parameters

	$\displaystyle Q_{20}$	$\displaystyle=\sqrt{\frac{5}{16\pi}}(2\langle z^{2}\rangle-\langle x^{2}\rangle-\langle y^{2}\rangle),$		(9)
	$\displaystyle Q_{22}$	$\displaystyle=\sqrt{\frac{15}{32\pi}}(\langle y^{2}\rangle-\langle x^{2}\rangle).$		(10)

Figure 6 displays the calculated $\beta$ of the ⁴He+2$n$+2$n$ configuration, showing that the value rapidly increases with increasing ⁴He-$2n$ distance. The mixing of such components could explain the large deformation of ⁸He discussed in the literature ($\beta=0.4$ [30]). The large experimental $\beta$ value is consistent with the large distance between ⁴He and $2n$s; however, it is not necessarily direct evidence of $2n$-$2n$ correlation with smaller opening angles. As shown in Fig. 6, large opening angles can also give large $\beta$ values.

To conclude, the four-neutron correlation has been a disputable topic in nuclear physics for decades. This study has investigated for the first time dineutron-dineutron ($2n$-$2n$) correlations in ⁸He using a reliable wave function obtained from a microscopic ${}^{4}{\rm He}+4n$ cluster model. To construct a ⁸He wave function, explicit two-$2n$ configurations were superposed along with many $4n$ configurations. The agreement between the theory and experiment is satisfactory. Overall, this study provides insights into the two-$2n$ correlation in ⁸He and its effects on the nuclear structure. The results show that a significant amount of the two-$2n$ components in the ground state of ⁸He could explain the strong nuclear deformation observed in recent studies. However, it should be noted that the strong deformation is not necessarily a direct evidence of $2n$-$2n$ correlations with small opening angles, as it can also be interpreted as configurations locating $2n$s at large distances.

Further experimental studies are needed to clarify the existence of the two-$2n$ correlations. Exploring the possibility of such $2n$-$2n$ correlations in other neutron-rich nuclei near the neutron dripline is also an interesting topic for the universal understanding of the emergence of the $2n$-$2n$ correlation.

This work was in part supported by JSPS KAKENHI Grants Nos. 18K03635, 22H01214, and 22K03618. The numerical calculations were performed using the computer facility of Yukawa Institute for Theoretical Physics, Kyoto University (Yukawa-21).